On proximal augmented Lagrangian based decomposition methods for dual block-angular convex composite programming problems

Abstract

We design inexact proximal augmented Lagrangian based decomposition methods for convex composite programming problems with dual block-angular structures. Our methods are particularly well suited for convex quadratic programming problems arising from stochastic programming models. The algorithmic framework is based on the application of the abstract inexact proximal ADMM framework developed in [Chen, Sun, Toh, Math. Prog. 161:237–270] to the dual of the target problem, as well as the application of the recently developed symmetric Gauss-Seidel decomposition theorem for solving a proximal multi-block convex composite quadratic programming problem. The key issues in our algorithmic design are firstly in designing appropriate proximal terms to decompose the computation of the dual variable blocks of the target problem to make the subproblems in each iteration easier to solve, and secondly to develop novel numerical schemes to solve the decomposed subproblems efficiently. Our inexact augmented Lagrangian based decomposition methods have guaranteed convergence. We present an application of the proposed algorithms to the doubly nonnegative relaxations of uncapacitated facility location problems, as well as to two-stage stochastic optimization problems. We conduct numerous numerical experiments to evaluate the performance of our method against state-of-the-art solvers such as Gurobi and MOSEK. Moreover, our proposed algorithms also compare favourably to the well-known progressive hedging algorithm of Rockafellar and Wets.

Appendices

Appendix 1: two-stage stochastic programming datasets

• assets. This network model represents the management of assets. Its nodes are asset categories and its arcs are transactions. The problem is to maximize the return of an investment from every stage with the balance of material at each node.

• env. This model assists the Canton of Geneva in planning its energy supply infrastructure and policies. The main objective is to minimize the installation cost of various types of energy, while meeting all the supply–demand at each node and satisfying several realistic constraints such as the environmental constraints.

• phone. This is a problem which models the service of providing private lines to telecommunication customers, often used by large corporations between business locations for high speed, private data transmission. The goal is to minimize the unserved requests, while staying within budget.

• AIRL. This dataset is used to schedule monthly airlift operations. The airlift operation is scheduled so that the restriction on each aircraft such as the number of flight hours available during the month can be satisfied. The cost such as the available flight time to go unused, switching aircraft from one route to another and buying commercial flights is the main objective to be minimized.

• 4node. This is a two stage network problem for scheduling cargo transportation. While the flight schedule is completely determined in stage one, the amounts of cargo to be shipped are uncertain and shall be determined in stage two.

• pltexp. This is a stochastic capacity expansion model that tries to allocate new production capacity across a set of plants so as to maximize profit subject to uncertain demand.

• storm. This is a two period freight scheduling problem described in Mulvey and Ruszczynski. In this model, routes are scheduled to satisfy a set of demands at stage 1, demands occur, and unmet demands are delivered at higher costs in stage 2 to account for shortcomings.

• gbd. This is the aircraft allocation problem where aircraft of different types are to be allocated to routes in a way that maximizes profit under uncertain demand, and minimizes the cost of operating the aircraft as well as costs associated with bumping passengers when the demand for seats outstrips the capacity.

Appendix 2: Derivation of solutions of (36) and (38)

We can rewrite the subproblem in (36) as follows:

$$\begin{aligned} \begin{aligned}&\min \limits _{z,y} \Big \{ -\langle b,\,y\rangle + \delta _{{\mathcal {K}}}^*(-z) + \frac{\sigma }{2}\Vert z+A^*y - \widehat{c}^k\Vert ^2 -\langle \delta ^{k},\,y\rangle \Big \}\\ =&\min \limits _{y}\left\{ -\langle b,\,y\rangle -\langle \delta ^k,\,y\rangle +\min \limits _{z}\left\{ \delta ^*_{{{\mathcal {K}}}}(-z)+\frac{\sigma }{2}\Vert z+A^*y-{\hat{c}}^k\Vert ^2\right\} \right\} \\ =&\min \limits _{y}\left\{ -\langle b,\,y\rangle -\langle \delta ^k,\,y\rangle +\sigma M_{\delta ^*_{{{\mathcal {K}}}}/\sigma }(A^*y-{\hat{c}}^k)\right\} , \end{aligned} \quad \end{aligned}$$

where the last equality comes from the definition of Moreau-Yosida envelope, and the equality holds for $z=-\mathop {\textrm{Prox}}_{\delta ^*_{{{\mathcal {K}}}}/\sigma }(A^*y-{\hat{c}}^k)$. By the property of Moreau-Yosida proximal mapping, $\mathrm {{Prox}}_{\delta ^*_{{{\mathcal {K}}}}/\sigma }(A^*y-{\hat{c}}^k)=(A^*y-{\hat{c}}^k)-\mathrm {{Prox}}_{\sigma \delta _{{{\mathcal {K}}}}}(\sigma (A^*y-{\hat{c}}^k))$, therefore, $z=\Pi _{{{\mathcal {K}}}}(\sigma (A^*y-{\hat{c}}^k))-(A^*y-{\hat{c}}^k)$, which gives (36). The derivation of (38) is similar, we omit the details.